Fractals, Markov

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

Liebovitch, L., Fischbarg, J., and Koniarek, J., Ion channel kinetics A model based on fractal scaling rather than multistate Markov processes, Mathematical Biosciences, Vol. 84, No. 1, 1987, pp. 37-68. 

The psychologist Maslow wrote that if the only tool you have is a hammer, you tend to treat everything as if it were a nail (4). Markov processes based on the Hodgkin-Huxley model had been widely used to describe ionic currents measured in many different experiments. However, in 1986, we began to use a new tool to analyze the patch clamp data. The insight gained from this new analysis has changed our ideas about the processes that open and close the ion channel. The new tool is based on fractals. 

The effective kinetic rate constant fceff is the probability for changing states when we observe the data at temporal resolution feff. Note that because 1 < d < 2, keff increases when we observe the channel at finer temporal resolution teff. That is, the faster we can look, the faster we see the channel open and close. If log fceff is plotted versus log eff, then eq 3 is a straight line. When the data are not fractal, this plot has other forms. For example, when there are only a few discrete states, such as those predicted by the Markov model, then there are a few well separated plateaus on this plot (10). Thus, without making any a priori assumptions about the data, we determine the function fceff(leff) and thus plot log keS versus log Ieff. The form of this plot can thus tell us the characteristics of the channel kinetics. 

Inspection of a large amount of published data suggests that some channels are best described by fractal scalings, some channels are best described by discrete single Markov states, and other channels show fractal behavior at some time scales and discrete-state Markov behavior at other time scales. 

The discovery of the fractal properties of the single-channel recordings now suggests a different picture of the physical properties of the ion channel protein than the foregoing three properties that were suggested by the Markov model. 

Calculations of the potential energy function of a large number of different globular proteins demonstrate that these proteins all have a very large number of shallow local energy minima (26). This analysis is consistent with the physical properties of ion channel proteins suggested by the fractal properties of the channel data and inconsistent with the few deep minima predicted by the Markov model. 

Thus, the biophysical studies demonstrate that globular proteins have (1) a very large number of conformational states corresponding to many shallow local minima in the potential energy function, (2) very broad continuous distributions of activation energies, and (3) time-dependent activation energy barriers. All these properties are consistent with the physical properties of ion channels derived from the fractal properties observed in the channel data and are inconsistent with the physical properties derived from the Markov model. 

Kienker, P. Matching Fractal Ion Channel Gating Models with Markov Models Preprint, 1989. 

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